MOORE. — MINIMUM GEOMETRY. 221 



way that length will be preserved and in fact this can be done in an 

 infinite number of ways. 4 If we put, 



u = x, v = y. 



we will have the minimum developable mapped on the plane of A. P. G. 

 in such a way that distance is preserved. In this depiction the genera- 

 tors of the developable are carried into the lines through F: the 

 imaginary circle at infinity into the line /. The cuspidal edge will be 

 transformed into the point F. This then differs from the develop- 

 ment of the ordinary developable on the euclidean plane since a whole 

 curve is transformed into a point. The transformation, however, 

 does everywhere preserve length. 



For lines analogous to geodesies on a minimum developable then 

 we can take the lines which by this depiction go into straight lines. 

 As we saw before this depiction can be made in an infinte number of 

 ways and therefore on a minimum developable there are an infinite 

 number of simply isomorphic geometries. The curves which we 

 shall take as pseudo geodesies are, 



au -j- bv = 1. 



The distance between two points (wi, Vi), (u 2 , 02) measured on one of 

 these lines is 



d = Hi V 2 — U 2 Vi. 



The angle between two lines 



«i u + 61 v = 1, 

 a% u + 62 v = 1, 



can then be defined as, 



/3 = aj)2 — a 2 b\, 



and will then be the exact dual of the distance between two points. 

 The area of the triangle (u h Vi), (u 2 , v 2 ), (u 3 , v 3 ) will be 



A = 



1 U\ Vi 



1 u 2 v 2 . 



1 «3 V3 



4 From the ordinary formula the curvature of a minimum developable i 

 indeterminate. However they can all be mapped on a minimum cone (point 

 sphere). If the curvature of a sphere be taken as the reciprocal of the radius , 

 the curvature of the point sphere is infinite. If then we say that applicable 

 surfaces have the same curvature, the curvature of a minimum developab le 

 is infinite. 



